Agricultural and Forest Meteorology, 55 ( 1991 ) 167-179 167 Elsevier Science Publishers B.V., Amsterdam

Momentum roughness and zero-plane displacement of ridge-furrow tilled soil*

K.J. Mclnnes, J.L. Heilman and R.W. Gesch Department o['Soil and Crop Sciences, Texas A&M University, College Station, TX 77843, USA

( Received 7 August 1990; revision accepted 6 October 1990 )

ABSTRACT

Mclnnes, K.J., Heilman, J.L. and Gesch, R.W., 1991. Momentum roughness and zero-plane displace- ment of ridge-furrow tilled soil. Agric. For. Meteorol., 55:167-179.

Estimation of heat and mass transfer between the atmosphere and the ground surface by flux- profile relationships may be influenced by the values chosen for zero-plane displacement d and momentum roughness Z0m. This study was conducted to determine the effect of wind direction on d and Z0m of a r idge-furrow tilled soil. Fifteen-minute average wind-speed and air-tempera- ture profiles, together with wind direction, measured over a 28-day period above a ridge-furrow tilled field near College Station, TX, were used to determine d and Zorn as a function of wind direction relative to ridge orientation. The difference in height h between the ridge tops and furrow bottoms was 0.22 m and the spacing between ridges was 1 m. A trade-off between d and ZOrn that did not appreciably alter the fit of theoretical to measured wind-speed profiles made separation o l d and ZOm difficult. NO strong relationship was found between d or Zorn and wind direction when both were allowed to vary and the best fits of theoretical to measured wind- speed profiles were found by a least sum of error squares technique. When Zorn was held constant and d allowed to vary, the best overall fit was with Zom/h=O.032 and d/h varying from about 0.091 to 1.2 when the wind direction was parallel and perpendicular to the ridges, respectively. When d was held constant and Zorn allowed to vary, the best overall fit was with d/h = 0.85 and Zom/h varying from 0.011 to 0.055 when the wind direction was parallel and perpendicular to the ridges, respectively. The implication to estimation of heat or mass transport between the atmosphere and the surface of a ridge-furrow tilled soil where d and ZOrn are required in the calculations is that wind direction should be considered.

I N T R O D U C T I O N

Cultivation that leaves ridges and furrows of some form is widely practised throughout the world. There is a need to predict the vertical fluxes of heat and mass from the soil surface to the atmosphere in many of these areas. Calcu- lations of these fluxes based on flux-profile relationships are influenced, in

*Technical Article TA-25771 from the Texas Agricultural Experiment Station, College Station, TX 77843, USA.

0168-1923/91/$03.50 © 1991 - - Elsevier Science Publishers B.V.

168 K.J. MclNNES ET AL.

part, by the values chosen for the displacement height d and momentum roughness Zorn which determine the shape of the wind-speed profiles. For ex- ample, the equation describing the aerodynamic resistance to transfer of mo- mentum between the soil surface and a given height, a resistance often equated to that for heat and mass transport, contains both d and Z0m (Monteith and Unsworth, 1990). The effective aerodynamic characteristics of ridge-furrow tilled soil change with the angle the wind makes with respect to the direction of the cultivation and both d and Z0m are probably affected by wind direction. When the wind blows perpendicular to the ridges, the ridges along with soil clods may act as bluff bodies. When the wind is parallel, instead of acting as bluff bodies, the ridges may simply displace the profile above that for a flat surface.

Studies to relate d or Zorn to the height h of the rotlghness elements on the surface have met with limited success. For crop-covered surfaces the ratio Zom/h has been found to be about 0.13 (Paeschke, 1937; Tanner and Pelton, 1960) while the ratio d/h has been reported to be about 0.64 (Cowan, 1968; Stanhill, 1969). Chamberlain (1968) estimated the ratio Zom/h from wave- like roughness elements, with the wind perpendicular to the elements, to range from 0.07 to 0.12, depending on the height and spacing I of the elements. A value of 0.4 for the ratio d/h was reported for one arrangement with h/l= O. 1. Rider (1957) reported values of the ratio d/h ranging from 0.4 to 0.6 for ridge-furrow tilled soil, but gave no information on the direction of the wind in the study or possible effect of wind direction. Other formulations involving more characteristics of the roughness elements have been derived for d and Z0m over surfaces with bluff obstacles (Lettau, 1969; Wooding et al., 1973); however, the accuracy of these formulations is low, and there is still no good substitute for the experimental determination of the parameters by wind-speed profile analysis (Brutsaert, 1982 ).

The purpose of this experiment was to determine, from wind-speed profile analysis, the effect of wind direction on d and Z0m of a ridge-furrow tilled field.

MATERIALS AND M E T H O D S

Field measurements

Cup anemometers (Model 12102D, R.M. Young Co., Traverse City, MI )a and ceramic wick thermocouple psychrometers (Gay, 1972) were installed for profile measurements in the center of a 25-ha ridge-furrow tilled field near College Station, TX, from November 11 through December 8, 1989. The sensors were located 0.40, 0.52, 0.67, 0.86, 1.11, 1.44, 1.86 and 2.40 m above

~Trade names and company names are included for the benefit of the reader. No endorsement is implied.

MOMENTUM ROUGHNESS AND ZERO-PLANE DISPLACEMENT, TILLED SOIL

0.62

g v N

0.22

0

i i . i

RIDGE FURROW RIDGE

169

Fig. 1. Locations of heat-transport anemometers.

the tops of the ridges. The difference in height h between the ridge top and furrow bottom was 0.22 m and the spacing between ridges was 1 m. The ridges were oriented 26°E of N and min imum fetch was in excess of 230 m in all directions. Average wind speeds, dry- and wet-bulb air temperatures, and 5 °-bin wind-direction frequency were recorded every 15 min, with a model CR7X data logger (Campbell Scientific, Logan, UT) a. Average wind direc- tions were calculated from the wind-direction frequencies.

In addition to the above measurements, wind speeds were measured be- tween and up to 0.40 m above the furrows with an array of 20 heat-transport anemometers constructed in our lab after the design of Kanemasu and Tan- ner ( 1968 ). Anemometers were placed above the ground surface as shown in Fig. 1.

Determination of d and ZOm

The method for determining d and Z0m was by minimizing the sum of squares of error sse between measured and theoretical wind speeds. In the surface sublayer, the sum of squares of the error between n theoretical and measured mean wind speeds may be expressed as

sse ~ (u i - (u./k){ln[ (z i -d)/zom] + ~'tm } )2 ( 1 ) i=1

where u~ is the measured wind speed at height z~, u. is the friction velocity, k is von Karman's constant, and ~Um is a correction for atmospheric stability. To determine s& with given d and Z0m values, ~Um must be known. For un- stable conditions we used the formulations given by Businger ( 1975 )"

~/m =21n[ ( 1 + x ) / 2 ] +In[ ( 1 + X 2 ) / 2 ] - 2 arctan (x) +zc/2 (2)

170 K.J. MclNNES ET AL.

X=Om' (3)

0m= [ 1 - 1 6 ( z - d ) / L ] - 0 2 5 (4)

L = -pCp T u3./(g H k) ( 5 )

H= [pCp k (z -d)u . /Oh]OT/Oz (6)

0h = [ 1 - 1 6 ( z - d ) / L ] - ° s (7)

u . = ( l / n ) ~ (u~k/{ ln[(z i -d) / zom]+ ~m}) (8) i=1

Here, L is the Monin-Obukov scaling length, g is the acceleration due to grav- ity, H is the sensible heat flux, T is the potential air temperature, and pcp is the volumetric heat capacity of air. Equations ( 2 ) - ( 8 ) were solved itera- tively to determine L and calculate a value of ~Urn that was then used to correct the theoretical mean-wind profile in eqn. 1 for instability. In this study, z was referenced to the bot tom of the furrow.

For stable conditions we used the same procedure as above except we sub- stituted the formulation of 0m (Thom, 1975), derived with the Richardson number Ri

0m =0h = (1--5Ri) -~ (9)

where

R i= (g/T) (OT/Oz)/(Ou/Oz) 2 (10)

but kept ~u,, as a function o f L (Businger, 1975 )

7 J m = 4 . 7 ( z - d ) / L (11)

The reason for using the formulation of 0m with Ri as the argument and not L is that the iteration to find ~m does not converge for stable conditions when Om = Oh = ( 1 Jr 4.7 ( Z-- d ) / L ). The air-temperature gradient used in calculat- ing Ri and H, and Ou/Oz used in calculating Ri were determined from the slopes of power functions of temperature and mean wind speed with height, respectively.

Several approaches to finding the min imum SSe as a function of both d and Z0m exist. Steams (1970) simultaneously solved OSSe/Od= 0 and OSSe/OZOm = 0 for d and Zorn. We chose instead three approaches that searched the d-zom-SSe surface for the min imum SSe value. In the first procedure, a rectangular grid of sse values for combinations of Z0r~ and d was produced, and then the com- bination producing the min imum SSe was found from this matrix. Second, values of Zorn were determined for d values from 0 to 0.5 m by 0.01-m incre- ments, and third, d values were determined for Z0m values from 1 to 50 mm by 1 -mm increments. When either d or Z0m was set constant in these latter two procedures, the value of the other that gave the min imum ss~ was determined

MOMENTUM ROUGHNESS AND ZERO-PLANE DISPLACEMENT, TILLED SOIL 171

by the golden-section search technique (Shoup, 1979 ). The reasons for these latter two procedures will be discussed later. For all three procedures, only profiles where the average wind speed at 0.40 m above the ridge top was greater than 1.0 m s- 1 were used. The total number of profiles used was 1859. Limits on the values of d and Zorn used in the golden-section search were 0 < d < 0.62 m and 0.001 < ZOrn < 220 mm, respectively.

RESULTS AND DISCUSSION

As pointed out by Stearns (1970) and shown here in Fig. 2, the surface defined by a matrix consisting of the sse between theoretical and measured wind-speed profiles with combinations of Z0m and d shows a trough with a base that has little relief along a d-zom curve. Figure 3A shows a contour plot of sse derived with a hypothetically measured wind-speed profile defined for the eight sample heights used in this experiment and with d = 0.1 m, u. = 0.3 m s - l , Z0m=0 .01 m and ~ - / m = 0 . Overlain on this contour plot are the corre- sponding values of u.. Although we have clipped Fig. 3, note from Fig. 2, which was derived with the same parameters, that the trough extends to d> 0.22 m, the height of the ridges in this study, and would have extended to d < 0 m had we not set the limit d > 0 m. A plot o f d against Zorn for the best fits to the profiles where d< 0.22 m is given in Fig. 3B. If overlain on Fig. 3A, most of the points in Fig. 3B fall within the first contour level. This is in part because the values of d and Z0m chosen for the profile from which Fig. 3A was generated were in the ranges of realistic values for this surface. Values of d> 0.22 m giving the best fit were commonly obtained and along with the

O

Fig. 2. Typical d-zom-SSc surface for this study. The surface shown is for sse < O. 3 m 2 s - 2.

1 "/2 K.J. McINNES ET AL.

0.20

0.15

0.10

0.05

E o

"O

0.20

0.15

0.10

0.05

U ,

0.25 0.30 0.35

:>:!

1

| : • ; [ ] t * ° °

!iil,- !li~'

I .t . : . ~ . . : . . • o , °

• •~, ° , t • ** 1_***= * • . • •

*= , , ; , | ~ = i t : : = = , : il'l

5 10 15 20 25 30

Z0m (rnm)

S S e

0.40 0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

Fig. 3. Contour plot ofsse between wind-speed profiles defined with combinations of ZOrn and d and a profile defined by zorn= 10 mm, d=0.1 m and u.=0.3 m s -~ (A). Relationship between d and Zorn from least error squares fitted wind-speed profiles (B).

corresponding value of Zorn produced an ss e that generally fell within the first contour level o f Fig. 3A.

From the trough in Fig. 2, it was evident that different combinations of d, u, and zorn gave nearly equivalent fits to wind profile data. We determined the relationships between d and Zorn from combinations of the two parameters that produced the least ss~ using the eight heights mentioned above while al- lowing u. to vary and trading d for Zorn (Fig. 4) . On this figure, vertical lines from a given ZOm intersect the curves at d values producing perfect fit. Hori- zontal lines from a given d intersect the curves at Zorn values producing perfect fit. Other combinations of d and Zorn falling on the same curve represent val- ues giving the least s s e fit to the profile o f best fit when either d or Zorn is changed. These curves are valid for combinat ions o f d and Zom when measure- ments are taken at the same sample heights as mentioned above. Curves sim- ilar to these could have been used to explain the diurnal variation in d and Zorn found by Pieri and Fuchs (1990) for a wide-row-spaced cotton crop. These

MOMENTUM ROUGHNESS AND ZERO-PLANE DISPLACEMENT, TILLED SOIL

0.4

°2: I I

1 10 60

Zom (mm)

Fig. 4. Least ss~ trade-off curves for d and Z0m.

173

trade-off curves may be the same for different profiles. Profiles defined from the combinat ion of d = 0 . 4 m and Z0m=2 mm, for example, share the same curve as profiles from the combination of d = 0 m and ZOrn = 30 mm. The points of best fit on the curve for different combinations, though, are obviously not the same. In practice, because the theoretical wind speeds in eqn. 1 are not an exact description of real wind speeds and because of experimental error, the point of best fit on the curve may not be as well defined as it is in theory. The friction velocity u. associated with these curves increases as d decreases or Z0m increases (Fig. 3A). An independent measure of u. may aid the choice of the best combinat ion of d and Z0m. However, if _+ 10% were a reasonable de- gree of accuracy for measurements of u., there would still be considerable uncertainty in d and ZOm (Fig. 3A).

The relationships between Z0m and wind direction and d and wind direction that resulted from our first procedure were not well defined (Fig. 5 ). An in- crease in ZOm as the wind shifted from parallel to perpendicular to the ridges is shown, but there was much scatter. Because we felt much of this scatter was the result of the trade-off between d and Zorn, we chose then to follow the sec- ond and third procedures to determine the effect of wind direction on d and ZOm. In the second procedure, we determined the value of Zorn that gave the best overall fit to the profiles, then held it constant and found how d varied with wind direction. Here, overall implies the sum of the sse for all 1859 pro- files. The value of Zorn that gave the best overall fit was 7 m m (Fig. 6 ). The overall sse when Zorn = 7 m m was 10.5 m 2 s-2, about 27% greater than the total of 8.3 m 2 s -2 for the combinations o f d and Zorn giving the best fits. Consid- ering there were 1859 profiles overall, the difference of 2.2 m ~ s -2 in fit pro- duced an added error that was well beyond the accuracy of our anemometers. Values of d at wind directions perpendicular to the ridges exceeded the heights

174 K.J . M c I N N E S E T A L .

A g

"0

E g

E ,4'

0.5

0.4

0.3

0.2

0.1

0.0

0.06

0.04

0.02

0.00

• .. ~ : - . F ~ ,~ ° . ° - -° °

..° ° . ° ~ ° . . . ~ **% o . , * • :

:;-:-":.:.;::- ;-.-;:--.:,. :,-;v ,'.

• ° ° ° *.° .~ . , °°-*. °

. - , : . . . • - . . . .

- . . . . . : . - . . . - . ~ . , = , - . - ~ , a , . . ~ : v . - . ,

30 60 90

WIND DIRECTION

Fig. 5. Relationships between d and wind direction (A) and Zorn and wind direction (B) from least squares fitted wind-speed profiles. Wind directions 0 ° and 90 ° are parallel and perpendic- ular to the direction of the ridges, respectively.

Pt.

0

u'J

,,¢ rv

0

90"

80'

70'

60'

50-

4 0 -

3O

2O

10

0

° .

5 10 15 20 25 30

Z o m (mm)

Fig. 6. Relationship between overall sse and Zorn for all profiles measured.

M O M E N T U M R O U G H N E S S A N D Z E R O - P L A N E D I S P L A C E M E N T , T I L L E D S O I L 175

of the ridges by about 0.04 m (d/h = 1.2 ) (Fig. 7 ). Values of d when the wind was parallel to the ridges were about 0.02 m (d/h = 0.091 ).

If values of d were to be restricted to being < h, then that fraction of d that would have been 0.22 m must be accounted for by a corresponding increase in Zorn (Fig. 4). For example, a value of Zom= 10 mm for all profiles gave values of d that were about equal to the height of the ridges when the wind was perpendicular to the rows and only increased overall sse by 0.9 m 2 s -2. When this value of Z0m was chosen, d varied from near 0 m when the wind blew parallel to the ridges to the height of the ridges when the wind blew perpendicular•

Our computer-generated contour map of the wind in and around the fur- row and ridges measured with the heat-transport anemometers (Fig. 8 ) sug- gests that a value of d> h may be warranted. In the wake of the furrow, an eddy extending above h formed when the wind was perpendicular to the ridge direction• This is consistent with Businger's ( 1975 ) analysis of wind profiles over shelter belts. When the wind flowed parallel to the ridges, the wind was horizontally uniform at a much lower height above the ridge tops than it was when the wind blew perpendicular. A value of d= 0 m when z is referenced to the bottom of the furrow is probably more objectionable than values of d> h, since the ridges must displace the profile compared with a fiat surface with the same roughness elements•

Our third approach was to find the value of d that gave the best overall fit to the data, hold it constant, and determine how Z0m varied with wind direc- tion. The value o fd tha t gave the best fit was 0.18 m ( Fig. 9 ). The least overall SSe with d constant was 12% greater than the total for the combinations of Zorn and d giving the best fits. The overall SSe with d= 0 m or d= h was not much greater than that with d= 0.18 m. With d= 0.18 m, ZOrn varied from about 2.5 to 12 m m ( Z o m / h from 0.011 to 0.055) when the wind was parallel and per-

0 . 4 -

0.3-

0 . 2 -

0 . 1

0 . 0

qD

- - 0 . 1 4 + 0 . 1 2 s i n ( ~ 0 / 9 0 - ~ / 2 )

' .;•-

I I 910 4 5

"~".: :.-.. ".'" "~4:

. . ' : i . .: . .,'=-~ I I I l I

1 3 5 1 8 0 2 2 5 2 7 0 3 1 5

W I N D D I R E C T I O N

3 6 0

Fig. 7. Relationship between d and wind direction 0 for Z0m = 7 mm. Wind directions 0, 180 and 360 ° are parallel to the ridges and 90 and 270 ° are perpendicular.

176 K.J. MclNNES ET AL.

v N

0.22

0.62

u (m s "1)

1.8 ]

1.5 I

1.2 I

0 . 9 1

0.6 I

0 .3 r

0 I

0.22

0

RIDGE FURROW RIDGE

Fig. 8. Contour plots of the wind speed near the surface of ridge-furrow tilled soil when the wind direction was perpendicular (A) and parallel (B) to the ridge direction. In A the wind was from right to left. The wind speeds at z=0.62 m for A and B were 1.6 and 1.7 m s -~, respectively.

,,=, n-

O

= E

rr

O

90-

80"

70"

60"

50"

40"

3 0 '

2 0 '

1 0

0 , I , 0.00 0.10 0.20

• • , i

0 " 3 0 0 . 4 0 0 . 5 0

d (m)

Fig. 9. Relationship between overall sse and d for all profiles measured.

MOMENTUM ROUGHNESS .AND ZERO-PLANE DISPLACEMENT, TILLED SOIL 177

pendicular to the rows, respectively (Fig. 10). Figure 10 may also be derived from either Fig. 5 or Fig. 7 using the trade-off curves. For example, from Fig. 7 when Z 0 m = 7 mm, d ranged from about 0.02 to 0.26 m when the wind was parallel and perpendicular to the ridge direction, respectively. If it were de- sired to find how Zorn varied when d= 0.18 m, the curves containing the inter- sections o f Zorn = 7 mm with d= 0.02 m a~d d= 0.26 m must be located on Fig. 4. Next, the Zorn values intersecting these curves at d= 0.18 m must be found. Visual estimation with this procedure produced Zorn values of about 2.5 mm when the wind was parallel to the ridges and 12 mm when the wind was per- pendicular. The solid line in Fig. 10 shows the directional variation in the d sine wave in Fig. 7 transformed into variation in ZOm with the above procedure.

The implication of these results, to estimation of heat and mass transport between the atmosphere and the surface of a ridge-furrow tilled soil when estimates of d and Z0m are needed, is that wind direction should be consid- ered. The same may be true for estimates of heat and mass transport when wide-row-spaced crops are considered. For example, the adiabatic aerodyn- amic resistance to momentum transfer rm, that is often used as a substitution for the resistances to heat and mass transfer, between the height d+ Zo~ and some reference height z with wind speed u is rm=ln[(z-d)/zom]2(k2u). When Zorn = 7 mm and z--2 m, the r~ that would be calculated when the wind direction is parallel to the ridges is 5% higher than the value when the wind is perpendicular. On the other hand, when d= 0.18 and Zorn varies, r m when the wind direction is parallel to the ridges is 72% higher than the value when the wind is perpendicular. From our data we cannot suggest which method is bet- ter. If one were to opt for holding Zorn constant and allowing d to vary so that rm remained nearly constant, the rather large and directionally dependent val-

20 - - F R O M FIGS. 4 AND 7

i s .~ ' : '~ • .

10 ~ " " " ~*""

5 ". ~ ' . , . . . , . . ¢~.. ~. , , ,

0 4t5 910 I I :~ I I

0 135 180 2 5 270 315 360

WIND DIRECTION

Fig. 10. Relationship between Zorn and wind direction for d = 0.18 m. Solid line was estimated from the sine wave in Fig. 7 and the trade-off curves in Fig. 4.

178 K.J. MclNNES ET AL.

ues of d would make estimation of a surface (height d + ZOm) temperature or gas density for the calculation of fluxes difficult.

CONCLUSIONS

Both d and Z0m of ridge-furrow tilled soil probably vary with wind direc- tion; however, because the trade-off between the two parameters does not radically alter the least error squares fit to the data, separating them in this study was not possible because of a lack of an independent measure of u.. Even with an independent measurement of u., separation may have been dif- ficult. When d was held constant and Zorn allowed to vary with wind direction the overall fit to the data was better than when Z0m was held constant and d allowed to vary. Estimates of r m varied more with wind direction when d was held constant and Z0m was allowed to vary than when Zorn was held constant and d allowed to vary. Detailed studies to determine the roughness lengths and displacement heights for heat and mass are needed if estimates of fluxes from ridge-furrow tilled soil using an aerodynamic resistance are to be performed.

REFERENCES

Brutsaert, W., 1982. Evaporation into the Atmosphere: Theory, History, and Application. D. Reidel, Dordrecht, 299 pp.

Businger, J.A., 1975. Aerodynamics of vegetated surfaces. In: D.A. de Vries and N.H. Afgan (Editors), Heat and Mass Transfer in the Biosphere. Wiley, New York, pp. 139-165.

Chamberlain, A.C., 1968. Transport of gases to and from surfaces with bluff and wave-like roughness elements. Q. J. R. Meteorol. Soc., 94:318-332.

Cowan, I.R., 1968. Mass, heat and momentum exchange between stands of plants and their atmospheric environment. Q. J. R. Meteorol. Soc., 94: 523-544.

Gay, L.W., 1972. On the construction and use of ceramic wick thermocouple psychrometers. In: R.W. Brown and B.P. van Haveren (Editors), Psychrometry in Water Relations Re- search. Utah Agricultural Experiment Station, Utah State University, Logan, UT, pp. 251- 258.

Kanemasu, E.T. and Tanner, C.B., 1968. A note on a heat transport anemometer. BioScience 18: 327-329.

Lettau, H., 1969. Note on aerodynamic roughness-parameter estimation on the basis of rough- ness element description. J. Appl. Meteorol., 8: 828-832.

Monteith, J.L. and Unsworth, M.H., 1990. Principles of Environmental Physics. Edward Ar- nold, New York, 291 pp.

Paeschke, W., 1937. Experimentelle Untersuchungen zum Rauhigkeits- und Stabilitgtsproblem in der bodennahen Luftschicht. Beitr. Freien Atmos., 24: 163-189.

Pieri, P. and Fuchs, M., 1990. Comparison of Bowen ratio and aerodynamic estimates of eva- potranspiration. Agric. For. Meteorol., 49: 243-256.

Rider, N.E., 1957. Water loss from various land surfaces. Q. J. R. Meteorol. Soc., 83: 181-193. Shoup, T.E., 1979. A Practical Guide to Computer Methods for Engineers. Prentice Hall, En-

glewood Cliffs, N J, 255 pp.

MOMENTUM ROUGHNESS AND ZERO-PLANE DISPLACEMENT, TILLED SOIL 179

Stanhill, G., 1969. A simple instrument for the field measurement of turbulent diffusion flux. J. Appl. Meteorol., 8:509-513.

Stearns, C.R., 1970. Determining surface roughness and displacement height. Boundary-Layer Meteorol., 1:102-111.

Tanner, C.B. and Pelton, W.L., 1960. Potential evapotranspiration estimates by the approxi- mate energy balance method of Penman. J. Geophys. Res., 65: 3391-3413.

Thorn, A.S., 1975. Momentum, mass and heat exchange of plant communities, p. 57-109. In: J.L. Monteith (Editor), Vegetation and the Atmosphere. Academic Press, New York, pp. 57-109.

Wooding, R.A., Bradley, E.F. and Marshall, J.K., 1973. Drag due to regular arrays of roughness elements of varying geometry. Boundary-Layer Meteorol., 5: 285-308.